


The Manhattan Problem

by Prochytes



Category: Black Panther (2018), Iron Man (Movies), Marvel Cinematic Universe, The Avengers (Marvel Movies)
Genre: Gen, Hurt/Comfort
Language: English
Status: Completed
Published: 2018-05-27
Updated: 2018-05-27
Packaged: 2019-05-14 13:10:05
Rating: Teen And Up Audiences
Warnings: Major Character Death
Chapters: 1
Words: 2,056
Publisher: archiveofourown.org
Story URL: https://archiveofourown.org/works/14770241
Author URL: https://archiveofourown.org/users/Prochytes/pseuds/Prochytes
Summary: Shuri, Tony, Nakia, and the mathematics of loss.





	The Manhattan Problem

**Author's Note:**

> Big spoilers for the 2018 _Black Panther_ film and _Avengers: Infinity War_.

There are voices, again. Processing them for content would waste resources. But voices there are, all the same.

“… folly to lay it on the shoulders of a child. Our borders gape open for the taking, and what does she do? She sits and stares.”

“M’Baku…”

“Wakanda’s neighbours know us for our true selves, now. We have no Heart-Shaped Herb, no…” The deep notes falter for a moment, before they are resumed. “How long before the jackals sniff at our door? She must lead.”

“She will.”

“Say on, Nakia. Trust your words to make it so. I do not.”

The retreat of heavy footsteps, and a crash. Silence, for a time.

“Add another murdered door to the butcher’s bill. M’Baku must be allowed his dramatic exit. Is it all a performance with him, Okoye? Or is it real?”

“It is always a performance with M’Baku. And always real. He lost his children, Nakia. Every one of them.”

“I…. I am sorry. But what can we do? A day now since she ate, or slept, or spoke. As M’Baku said, she sits and stares.”

“We wait, then. And hope that Bast will guide her home.”

***

n!/⌊n/2⌋!⌈n/2⌉!

***

“Hello, Shuri. We haven’t been introduced. My name is Tony Stark. I’m smarter than you.”

A pause.

“… aaaand the appeal to vanity is a bust. That play would have worked on me. That play _did_ work on me, once or twice, when I’d thought myself to where you are, now. So this is what that looks like, from outside. No wonder Pep… no wonder people always freaked out when they saw me do it.”

Another, longer pause.

“Your friends don’t understand. I never told my folks, either. Mom never found out and Howard… Howard had been there himself. They think you’re inert; I know you’re _working_. Too busy to be hassled with weak shit like incoming sense data, or non-autonomic motor functions.

“I’m one of about ten people, maybe, who can imagine the cathedrals you’re building, the dragons you’re slaying, in there. But take it from a guy who’s courted addiction in all its gorgeous guises: thought can be a drug, like any other. Maybe I could out-think all this shit by myself, if there were time. But Time is someone else’s knuckle-duster, now. We need you here, Shuri. Your people need you. Please?”

Silence sliding back upon itself.

***

n!/⌊n/2⌋!⌈n/2⌉!

***

“Mr. Stark?”

“That’s what they usually put on the rap-sheet. Nakia, isn’t it?”

“Yes. Okoye said that I would find you here, with her.”

“Yeah. I was careful to clear this royal visit with Bald and Frowny first. Never knew that Fury had a sister.”

“Pardon?”

“Joke. Not a good one. What’s up with that scrap of paper?”

“This was in front of Shuri, when we found her.”

“Her hand-writing?”

“Yes. I hoped that this would explain why she is…. how she is. But it’s just some equation – all ‘n’s and ‘2’s and exclamation marks. I’m not an expert at math. They say that you’re almost as smart as Shuri…”

“Jeez, you Wakandans know how to hit where it hurts…”

“… so I wondered whether you could tell me what it meant.”

“Let me see.” A rustle. “Ah. Right. I… I don’t know the value of _n_ , Nakia. Sorry. I… I could use a drink right now. What does this palace have in the way of Scotch?”

***

n!/⌊n/2⌋!⌈n/2⌉!

***

“I see you found the Scotch.”

“I did.”

“You might want to put that tumbler down.”

“Why?”

Glass shattering on stone, and then a gasp.

“I’m not an expert at math. I _am_ an expert at liars. You’re a damned liar, Tony Stark. One and half more pounds of pressure on this lock will make you an unconscious one. What is _n_?”

“I didn’t lie… exactly. I don’t know the _value_ of this _n_. No one does, including Shuri. She isn’t trying to solve that equation – without the value of _n_ , it’s insoluble. But she fled inside her head from what that equation means.”

“What’s she doing?”

“Working. Solving problems. Anything, everything, so as not to have to think about what’s on that paper. _A strong fortress is our mathematics._ Shuri’s lowered the portcullis and raised the drawbridge.”

“Sounds like you’ve been there.”

“I have. It’s beautiful, and productive. But, at heart, it’s a shell game you play with pain. ‘Distracted from distraction by distraction,’ Pepper would say. She always did read more than me.”

“Pepper?”

“Pepper Potts. My bride. She… she didn’t make it. A world without her is no world at all.”

“I’m sorry, Mr. Stark. And sorry, too, for the violence. I overreacted.”

“Hey – what’s a loss of circulation between friends? I kinda earned it.”

“You still haven’t told me what the equation means.”

“You sure you want to go there, Nakia? When it comes to mansplaining, you’re looking at the undefeated reigning eleven times champion of the world. And I suck at exposition to normal folks. No offence, but you all think so goddamned _slowly_. I never know whether to stick with a ‘tectonic’ pace or gamble on a sprightly ‘glacial’.”

“Shuri always says the same. The only person she could explain her thoughts to was… Hmm. Tell me about the equation, Mr. Stark. Unfold it for me. I think… I think that that might help.”

“What? Ah – yes. Well played, Nakia – very well played, indeed. I never thought I’d be the _third_ smartest in a room.”

***

n!/⌊n/2⌋!⌈n/2⌉!

***

“It’s basic, really. Kids’ stuff. Actual kids, I mean. Not ‘built his first A. I. when he was seven’ here, or ‘developed vibranium tech when she was a regal foetus’ over there. It’s a version of The Manhattan Problem.”

“What’s that?”

“Picture a grid of streets. Let’s say the grid is five blocks north to south, and an equal number east to west.”

“A square made up of twenty-five smaller squares.”

“That’s right. Imagine that you start as far north-west as you could be – the top left corner. You choose a path through all those streets, aiming for the bottom right-hand corner, the extreme south-east. Whenever you reach a junction – the corner to one of those twenty-five small squares – you have to make a decision, unless you’ve already reached the other sides of the big one: do I go south, or do I go east? You can’t go west; you can’t go north. You can’t go back. You can never go back.”

“Mr. Stark?”

“Sorry. I was… I was distracted. Anyway, here’s the question: how many different routes can you take from the north-west corner of the grid to the south-east one, if you stick to the lines, and you can only ever move south or east?”

“Is the grid-plan what makes this ‘The Manhattan Problem’? Manhattan’s not that small, and not a square.”

“Good points, both. But that’s the traditional name, so let’s roll with it. Always bad business to ignore a trademark.”

“How does one solve it, then, this Manhattan Problem?”

“With the constraints I’ve described, you’ll be making ten moves to reach one corner from the other: no more, no less. Five moves east, and five moves south. As long as your journey includes five of each – doesn’t matter which order, as long as there’s five of one and five of the other – you’ll end up at the far corner. So, you’ve got ten moves, and have to pick five out of those ten to be the eastward ones.”

“Why does that help?”

“That gives us another way to handle the problem. Navigating our bonsai NYC becomes like drawing cards: five cards from a deck of ten, if you can’t put the cards back, and the order you pick them doesn’t matter. The sum of the individual ways you can do _that_ is the solution to our Manhattan Problem. How many different ways could you pick one card from a deck of ten?”

“Ten.”

“If you picked again without putting that first card back, how many possible results for that second choice?”

“Nine.”

“Very good. Each of those first ten possibilities produces another nine. If you go on picking, each of those ninety possibilities generates another eight; then each of those seven hundred twenty possibilities births another seven, and so on. The number of different ways you can pick ten cards from a deck of ten, if the order matters, and you can’t put cards back, is ten times nine times eight times seven times six… and so on, all the way down to one. That’s ten factorial, or ten with an exclamation mark, because ten is surprised at how quickly that escalated. Factorials get real big, real fast.”

“But we’re only drawing five cards, and the order doesn’t matter."

“Exactly. The last five picks aren’t really there. We don’t need the five times four times three times two times one at the end, for what we’re doing, so we divide ten factorial by five factorial, to remove that set of possibilities. We also don’t need all the versions of our five selected cards which are just the same cards in a different order. How many ways, based on what I’ve said, can you arrange five cards from a deck of five without redraws, if the order matters?”

“Five factorial, once more. Do we use this to divide again?”

“We do. That takes away the order issue. If you divide ten factorial by five factorial, and then divide the result by five factorial again, you get two hundred fifty-two. Two hundred fifty-two possible hands; two hundred fifty-two possible routes. That trick isn’t limited to a grid five squares by five. For any square grid, double the number of blocks across, and call that number _n_. Make _n_ factorial; divide it by the factorial of the number of blocks across – _n_ over two, now - and then divide the result by that second factorial once more. _Voilà_ : you just solved The Manhattan Problem.”

“I see – I think. That’s very clever.”

“It’s beautiful. Wouldn’t be worth jack shit in the genuine Manhattan, though. What’s left of Manhattan, that is – I kinda just broke the city. Again. The only kid I knew in NYC, he wouldn’t have been sticking to the grid. He’d have been swinging twenty stories above it, whooping his ass off.”

“Sounds like quite a kid.”

“He was. His name… his name was Peter. He died afraid, hoping that I could save him.”

Another silence.

“We’re not talking about a route through city streets, are we, Mr. Stark?”

“No. We’re not.”

“We’re talking about how many different ways you could select half from any given number of things.”

“Yes.”

“And leave the rest.”

“Yes.”

“The _n_ which you and Shuri don’t want to think about, it is… it _was_ very big, wasn’t it?”

“Yes. It was.”

“I think I need that Scotch.”

“I think I’ll join you.”

***

n!/⌊n/2⌋!⌈n/2⌉!

***

“There’s a related result. One I always loved, when I was a kid. My guess would be that Shuri loved it, too. Now – maybe – not so much.”

“What was this result?”

“Remember our pick of five cards from a deck of ten, with no redraws, and not caring about the order? Well, the number of ways you can pick five cards from ten like that will equal the number of ways you can pick five cards from nine, plus the number of ways you can pick four cards from nine. You don’t need any fancy math to prove that, either.”

“How so?”

“Think about any one of those ten cards. Either it gets picked, or it doesn’t. If it doesn’t, you still need five draws from the other nine. If it does, you need four draws from the same pool. See? That applies to every single card. The friendly neighbourhood Jack. The red-haired Queen. The beloved King.”

“Mr. Stark…”

“I tried to save him, Nakia. I tried to save them all. I wasn’t smart enough.”

“Only smart enough to save so very many. This is a time for tears: it is good, at last, to see you weep. It is not a time for blame.”

“Huh? I’m not weep… Oh. I see. I… I’ll leave you two alone, to talk.”

“Stay, Tony Stark. You and Shuri have much to tell each other.”

FINIS

**Author's Note:**

> Tony remembers Pepper quoting from "Burnt Norton", by T. S. Eliot. The aphorism "A strong fortress is our mathematics" is attributed to Pál Turán. For the purposes of his exposition, Tony simplifies the equation a little: he does not explain the floor and ceiling functions which accommodate an odd value of _n_. There are also other, slightly different Manhattan Problems. 
> 
> The fic is deeply indebted to _Combinatorics: A Very Short Introduction_ by Robin Wilson (Oxford, 2016), and to arachnekallisti, for checking over the mathematics.


End file.
